Aircraft: B737
Phase of flight: Cruise
Anomaly: Aircraft Equipment Problem Less Severe; Deviation - Altitude Excursion From Assigned Altitude; Deviation - Track / Heading All Types; Deviation / Discrepancy - Procedural Clearance; Inflight Event / Encounter Weather / Turbulence

Before leaving the hotel; I tried to download the flight plan and route. I was unsuccessful as there was no paperwork loaded in Sabre. The CA was also trying on the way to the airport but could not get a data/internet connection on his iPad. We discussed this on the van and decided we would ask ops to print a copy. At some point; he was locked out of his iPad and needed to have it reset by IT. The CA requested a printed copy of the paperwork from ops personnel. I tried unsuccessfully to download the flight plan. I also was unable to obtain a data connection through any of the carriers available to my iPad in MPTO. Once on the airplane; the CA used a phone from our maintenance technician to have his iPad unlocked and was able to download the flight plan and Jeppesen Pro route. I continued to set up the cockpit for our departure as the CA spoke with the flight attendants. I was still not getting any data connection so the CA air dropped the flight plan and paperwork as well as the Jeppesen pro data/route to my iPad. At some point prior to this; station personnel delivered one copy of the flight paperwork to the aircraft. I used this to enter into the FMC and obtain preliminary takeoff data. The CA briefed the departure and we ran the appropriate checklists. The next hour of the flight was ops normal with just some light chop on the climb. Once at cruise for 20-30 mins; we turned off the seat belt sign and I made an announcement about our time enroute; keeping the aisles clear for the flight attendants and keeping your seatbelt on at cruise even when the sign is off in the event we encounter any unforecasted turbulence. A short time later; we encountered some light chop so we turned the seatbelt sign back on and an announcement was made. We began deviations to the east just prior to SIGMA. We were in mostly clear air with some high thin cirrus and very light chop at this point. I had my radar on 160 miles and the CA was on 80 miles. We could see weather on both sides of us about 20-30 or so miles either side. No red; mostly green and a few yellow returns. At some point; Merida Center cleared us direct KELHI once we were done deviating. The weather was turning further east as we progressed north and looked like it may be closing so we decided to head back towards KELHI. There appeared to be an opening in the weather on both the radar and visually out the window as we saw blue sky. We decided this would be the safest route to pass by the weather so we proceeded in that direction. We called the flight attendants and asked them to be seated and made a PA to the passengers to remain seated. I also heard the FA's making an announcement which I assume was in Spanish asking the passengers to remain seated. Although we asked the flight attendants to remain seated; I heard noises from the galley and what sounded like the bathroom door opening and closing. I almost said something to the Captain but their noncompliance with us asking them to remain seated happens so often that I didn't. As we continued our left turn; there was nothing on the radar and although we were IMC; it was very thin and it appeared visually that there were no storms in front or to the sides of us. A short time into this; we encountered severe turbulence. The aircraft rolled right and we lost approximately 400 feet of altitude. The autopilot turned off; there was some buffeting and the aural autopilot disconnect warning sounded as well as the 'bank angle' warning and altitude chime. The Captain took manual control of the aircraft and rolled the wings level and stopped the descent. We stabilized the aircraft; turned both ignitors on and reengaged the autopilot. I made a PA to the flight attendants to be seated immediately. After the CA and I talked about the current aircraft state; autopilot on; roll and vertical mode selected on the MCP and he called back to check on the flight attendants. We asked them to check on the passengers and report back to us. At this point; we were heading direct to KELHI. The FAs told us there were 3 passengers and one FA with suspected injuries. There was a Doctor on board who assisted in assessing the passengers. I sent an ACARS to dispatch advising them of the turbulence encounter. I took PF duties and ATC communication and the CA attempted to get a phone patch to dispatch and medlink. I advised Merida center of the turbulence. I relayed some additional info to dispatch through ACARS. The CA was having a difficult time communicating on ARINC. The Doctor on board and the one on Medlink didn't indicate a diversion was needed. The Captain and I discussed and proceeded on. Once past KELHI and in contact with Houston Center; I advised them of the turbulence encounter and asked for priority handling into ZZZ. They immediately cleared us direct and we continued. The CA and I discussed and complied with all the items listed in the QRH Turbulence guide and entered the pertinent codes into ACARS for a Maintenance write up. We landed in ZZZ and taxied to the gate where we were met by CFR and representatives from [company]. [Report narrative contained no additional information.] B737 flight crew reported severe turbulence with injuries 80 NM east of MMUN at FL360.